具有n次方非线性恢复力的系统强迫振动中的次谐波响应

具有n次方非线性恢复力的系统,在强迫振动中可能出现频率为激振频率ω的1/n的次谐波响应.但实际为人们所发现的只是一些特殊系统,例如n=3(杜芬方程)的情形,而缺乏一般性的理论证明.本文利用基本摄动法,从理论上就n为任意奇数值(对于恢复力而言,n只可能为奇数)的情形,讨论了次谐波响应的可能性,对上述结论给出一般性的理论证明,并提出求次谐波共振解的一般较简便的方法.

Subharmonics in the Forced Vibration of a System with a Nonlinear Restoring Force Proportional to nth Power of Its Displacement

The subharmonics of order n may appear in a vibrating system with a nonlinear restoring force proportional to nth power of its displacement. But this phenomenon is found only in some special systems, for instance, in the Duffing equation, and the general theoretical verification has not be given yet. In this paper, the theorem is proved to the hilt by using the fundamental perturbation method and a simple method is developed for calculating the subharmonic resonance.